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  • 标题:PRECAUTION AND LIQUIDITY IN THE DEMAND FOR HOUSING.
  • 作者:BALVERS, RONALD J. ; SZERB, LASZLO
  • 期刊名称:Economic Inquiry
  • 印刷版ISSN:0095-2583
  • 出版年度:2000
  • 期号:April
  • 语种:English
  • 出版社:Western Economic Association International
  • 关键词:Fixed assets;Housing;Liquidity (Finance);Mortgages

PRECAUTION AND LIQUIDITY IN THE DEMAND FOR HOUSING.


BALVERS, RONALD J. ; SZERB, LASZLO


We exploit cross-sectional mortgage data to investigate the importance of liquidity constraints and a precautionary motive in the demand for housing. Households that are not liquidity constrained consume housing services essentially as the life cycle hypothesis suggests but with a significant precautionary component. Households that are liquidity constrained, in terms of not meeting standard loan-to-value or payments-to-income constraints, are similar to unconstrained households in most respects, including the precautionary motive, but they respond somewhat less to fluctuations in their lifetime income--suggesting some influence of bank-induced liquidity constraints. We additionally find, however, that banks enforce liquidity constraints only weakly. (JEL D91, D12, R21)

I. INTRODUCTION

The benchmark theory in the economics of consumption is the life cycle or permanent income hypothesis (LCPI) based on the work of Modigliani and Brumberg [1954] and Friedman [1957]. Its empirical validity is sometimes questionable, especially when compared against alternatives that incorporate liquidity constraints or the precautionary motive for saving. See, for instance, Deaton [1992] for a survey. Rejection of the LCPI may, however, be just a manifestation of the poor quality of consumption data. At the micro level, these data are typically subject to the reporting biases and omissions inherent in consumer surveys. There are further difficulties in measuring a desired stream of consumption services: nondurable consumption goods are a small fraction of total consumption and, in the case of food essentials, hardly a consumer choice variable; durable consumption goods have the feature that they are lumpy and that the consumption services at any point in time are hard to measure.

We propose to test the LCPI at the micro level with a quite different type of data. The data are obtained from the Residential Mortgage Finance Database collected by the National Association of Realtors. They are based on actual home purchases and the associated mortgage transactions reported by realtors and thus avoid some of the reporting biases of the consumer surveys. More important, they allow us to accurately identify households that, in principle, have instantaneous access to additional liquidity by lowering the down payment for the house purchase or increasing the size of the loan. Further, the purchase of a house provides an accurate measure of a relatively large fraction of consumption services obtained for a medium to long horizon and measured at the time of the consumption decision. Although our data also have some serious shortcomings, they allow a look at the LCPI from an unconventional vantage point.

Perhaps the most influential study employing the standard consumer survey in testing the LCPI at the micro level is that of Hall and Mishkin [1982]. It examines food expenditure from the Panel Study of Income Dynamics. One of their key results is that consumption tracks income more closely than would be expected under the LCPI. They demonstrate that this result could be explained by assuming a group of liquidity-constrained consumers, consuming their income at each point in time, of 20% of the total sample. Some of the drawbacks of this study are the use of food expenditures to represent consumption, the underreporting of income, and absence of proper wealth data inherent in the consumer survey, and the associated inability to individually separate liquidity-constrained consumers from the rest of the sample.

Several previous studies have attempted to improve on this way of testing the LCPI by examining other data sets. Bernanke [1984] used data on durable consumption goods (automobiles) to overcome the drawbacks of using food consumption in the Hall and Mishkin study. His results support the LCPI but are still suspect because of possible reporting biases and the lack of good wealth and savings observations. Hayashi [1985] employs the Survey of Financial Characteristics of Consumers conducted by the Board of Governors of the Federal Reserve System, which includes detailed information about income and wealth variables. These data enable him to identify as consumers who are not liquidity constrained those who save a lot at the current time. In these data, the constrained households consumed less than predicted by the results for the whole sample, suggesting that borrowing constraints were effective, thus providing evidence against the LCPI. As noted by Deaton [1992, 155], Hayashi's data do not contain consumption, so it must be inferred from disposable income and asset transactions. In addition to being susceptible to reporting biases, the procedure also misrepresents the service flow of durable consumption by measuring it as the total amount spent on durable consumption goods (this would be particularly inappropriate if a house was bought!).

Jones [1990] is similar to our study in the sense of using housing demand to test the LCPI. His hypothesis is that liquidity-constrained households should base their housing demand mostly on current net wealth whereas nonconstrained households should also be sensitive to their current labor income, as representative of lifetime earnings. The study is conducted based on the young households (head of household age up to 34 years) in the Canada Survey of Consumer Finances. This data set provides good information about the households' asset positions but has some drawbacks in not providing the purchase date and price of the house and using reported market value for the price of the house.

Jones's results show that net wealth is more important than income in explaining the housing demand of these young Canadian households. His results thus support the importance of liquidity constraints as contrasted against the pure LCPI hypothesis. Note, however, that Jones's basic interpretation of the LCPI does not include a precautionary motive in the determination of consumption. [1]

The shortcomings of our data will be discussed in section III, but the main advantages are that the surveys are completed by realtors based on actual transactions, that housing prices are observed at the exact time of purchase, that the dowupayment is reported without bias and may serve as an adequate measure of initial wealth, and that precise identification of nonconstrained households is possible.

Our data allow us to check by maximum likelihood corrected for truncation bias whether the permanent income elasticity equals one, as suggested by the life cycle hypothesis; whether a precautionary savings motive is identifiable; whether liquidity constraints matter to households or banks; and whether different groups of borrowers have different consumption patterns.

II. THEORY

We employ a basic model of the LCPI augmented with a precautionary savings motive. Xu [1995] decomposed the precautionary motive into one part related to income uncertainty and one part related to liquidity considerations. As we are able to identify households without current liquidity problems, and, likely, also without anticipated future liquidity problems, we can theoretically focus on the precautionary motive as related to income uncertainty only.

To derive a simple empirically testable equation in the absence of liquidity constraints the most crucial simplifying assumptions are the following: (1) the real rate of interest is equal to the rate of time preference; (2) the elasticity of substitution between housing services and other consumption services is constant; (3) uncertainty about income takes the form of a one-time shock realized at some time in the future. Consider then the life cycle consumption problem of a typical household:

(1) [Max.sub.[[{[H.sub.s], [C.sub.s]}.sup.T-1].sub.0] E[[[[sum].sup.T-1].sub.j=0] [[lgroup][frac{1}{1 + [rho]}][rgroup].sup.j] U([Z.sub.j])].

(2) s.t.

[Z.sub.t] = [[[[alpha].sup.[frac{1}{[sigma]}]] [[H.sup.[frac{[sigma] - 1}{[sigma]}]].sub.t] + [(1 - [alpha]).sup.[frac{1}{[sigma]}]] [[C.sup.[frac{[sigma] - 1}{[sigma]}]].sub.t]].sup.[frac{[sigma]}{[sigma] - 1}]].

(3) [W.sub.t+1] = (1 + r)[W.sub.t] + [Y.sub.t] - [Z.sub.t].

(4) [Z.sub.t] = [C.sub.t] + (r + [delta]) [[p.sup.H].sub.t][H.sub.t].

Equation (1) represents the standard time-separable utility function with finite lifetime. Expectations are taken conditional on information at time zero; life duration, T, differs among household heads depending on current age. The consumption index at time t, [Z.sub.t], is a constant elasticity by substitution (CES) function of housing consumption, [H.sub.t], and other consumption services, [C.sub.t], with elasticity of substitution equal to [sigma] as given in equation (2). Real wealth (with consumption services as the numeraire), [W.sub.t], evolves according to equation (3), with real interest rate r and stochastic real income, [Y.sub.t]. From basic duality theory real expenditure can be written as [Z.sub.t] and is given by equation (4) as the quantity of consumption plus the relative price of housing times the measure of housing quantity.

The relative price of housing is given as the Jorgensonian user cost (r + [delta])[[p.sup.H].sub.t] per unit of housing, which is similar for home-owners as well as renters (we ignore tax effects), as the real interest cost plus upkeep (depreciation) times the unit price of the house in real terms. Actual costs for the owner equal nominal interest minus the appreciation of the house in nominal terms, which equals the real interest rate if the housing market provides a proper hedge against inflation. If the house appreciates in real value terms, this should be incorporated in the [delta] term and may potentially lead to a negative [delta].

The uncertainty concerning future income is assumed to take the following simple form in order to model a precautionary motive for saving with relatively few additional complications

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [eta] [greater than] 0 and E([eta]) = 1. [2] The income uncertainty is resolved completely at future time s. The household making decisions at time 0 thus faces lifetime income uncertainty that is rationally anticipated to be resolved at (and no sooner or later than) a specific time s [greater than] 0. Since by assumption we have r = [rho], it follows straightforwardly that [3]

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The expressions for Z--the consumption level before the income uncertainty is revealed--and Q([eta])--the fraction of additional consumption after the income uncertainty is revealed--will be examined in the following. Solving the current budget constraint in equation (3) using the fact that [W.sub.T+1]. = 0 to obtain the lifetime budget constraint, and making use of equation (6) yields:

(7) Z [lgroup]1 + [frac{Q([eta])[[(1 + r).sup.T-s] - 1]}{[(1 + r).sup.T] - 1}][rgroup]

= PI([eta]),

where the left-hand side of equation (7) represent the per-period average present value of consumption expenditure from current time until death at time T and the right-hand side represents Deaton's [1992] formalization of the concept of permanent income, expressed in our framework as [4]

(8) PI([eta]) = [frac{r[(1 + r).sup.T]}{[(1 + r).sup.T] - 1}]

X [lgroup][W.sub.0] + [[[sum].sup.s].sub.j=1] [frac{Y}{[(1 + r).sup.j]}] + [[[sum].sup.R].sub.j=s+1] [frac{[eta]Y}{[(1 + r).sup.j]}][rgroup].

Optimal housing demand can be obtained from a two-stage budgeting process. First,

(9) [H.sub.t] = [alpha][([[p.sup.H].sub.t]).sup.-[sigma]] [Z.sub.t].

Taking expectations in equation (7), and using equation (9) together with equation (6) for t = 0, yields the demand for housing at time 0

(10) [H.sub.0] = [frac{[alpha]}{[(r + [delta]).sup.[sigma]]}][([[p.sup.H].sub.0]).sup.-[sigma]] E[PI([eta])]

x [[lgroup]1 + [frac{E[Q([eta])][[(1 + r).sup.T-s] - 1]}{[(1 + r).sup.T] - 1}][rgroup].sup.-1].

We next analyze the term in large parentheses in equation (10). The first-order conditions imply that consumption smoothing is attempted even for the period during which uncertainty is resolved. In particular,

(11) U'(Z) = E(U'{Z[1 + Q([eta])]}),

which determines the distribution of Q([eta]) together with equations (7) and (8).

The coefficient of variation of permanent income, [gamma] [equiv] [[sigma].sub.PI]/E[PI([eta])] may be obtained from equation (8). The appendix shows that [gamma] affects E[Q([eta])] or the term in parentheses in equation (10) positively given a "prudent" consumer as clarified by Kimball [1990]; if [omega] indicates the degree of prudence of the consumer, then an increase in [omega] also affects E[Q([eta])] positively. This term is further affected by T and E[PI([eta])] but the effect of either is ambiguous. The appendix derives, however, the effects for a constant relative risk aversion utility function, which yields

(12) 1 + [frac{E[Q([eta])][[(1 + r).sup.T-s] - 1]}{[(1 + r).sup.T] - 1}]

= q([gamma], [omega], T),

with partial derivatives, [q.sub.[gamma]] [greater than] 0, [q.sub.[omega]] [greater than] 0, and [q.sub.T] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [q.sub.E(PI)] = 0 and r and s are treated as fixed parameters. These results are less specific than those in Caballero's [1991] elegant model but more desirable empirically, as they do not rely on the implausible case of constant absolute risk aversion.

III. FROM THEORY TO MEASUREMENT

Based on the above model, we will investigate the determinants of the demand for housing. Before we proceed to the estimation, however, we will first describe the data and, subsequently present the model in a form that is estimable from these data.

Data Description

The data available to us from the Residential Mortgage Finance Database for the years 1988, 1989, and 1990 were collected by the National Association of Realtors through surveys completed by realtors in large metropolitan areas. These data consist of information from mortgage applications and other sources available to the realtors at the time of the closing of the purchase of a primary home. Variables included are purchase price of the house, down payment, loan amount, mortgage rate and type of mortgage, family size, borrower income, and age.

The total numbers of observations in the data set are 1,228, 1,138, and 922 for 1988, 1989, and 1990, respectively. All three years in our data set are similar in terms of representing low and stable inflation. We thus combine the sample points from all three of these periods. Sample points containing second mortgages, buy-downs, buyers over 65 years old, no or negative down payment, missing data, or obviously incorrect information are excluded. This leaves us with a total sample of 2,364 observations.

The data have several drawbacks. (i) They are not a panel and thus provide only current income information. (ii) The data provide limited information about nonfinancial borrower characteristics. Information about occupation, race, and education is not available, and we rely on age, and family size (as in Jones [1990]), and the unexplained component of the choice of mortgage type, to control for individual household taste variation. (iii) No information about income components and wealth is available. Nonhuman wealth must be measured by the down payment, which is certainly a questionable approximation but at least one that is based on an actual market transaction rather than obtained by household survey. And (iv) all households in our sample are homeowners, posing a sample selection problem that is difficult to correct.

On the other hand, the data offer several advantages. (i) They include diminished reporting bias due to provision by professionals based on recent market transactions; (ii) timeliness of the observations in terms of measuring housing consumption at the time of the transaction so that inertia has not yet contaminated the reliability of the observation as a measure of desired consumption (see Grossman and Laroque [1990] on the importance of inertia in the demand for illiquid assets); (iii) our data are homogeneous, consisting entirely of homeowners in large metropolitan areas during a relatively noninflationary period; and (iv) most important, our data allow identification of nonconstrained borrowers as those who, currently, could have obtained a larger loan from the bank or could have lowered their down payment because they face no loan-to-value or payment-to-income constraint. In addition, our introduction of these data to the study of consumption provides a new perspective.

Another data source, the Mortgage Interest Rate Survey (MIRS), is provided by the Office of Thrift Supervision. This data set provides monthly average regional house prices, which are used to construct our housing price index variable. Data on actuarial life expectancy, average taxes for the income brackets, and inflation rates are drawn from various issues of the Statistical Abstracts of the United States (years 1990-93).

Empirical Specification

In the following, we take in all cases time 0 to be the time of purchase of the house and omit the time subscript. To obtain the empirical model as it applies to individual take logs in equation (10) after substituting in equation (12). Rearranging to replace housing units for individual i, [H.sub.i], by the observable purchase price of the house, [[p.sup.H].sub.i][H.sub.i], yields

(13) ln [[p.sup.H].sub.i] [H.sub.i]

= ln[[(r + [[delta].sub.i]).sup.-[[sigma].sub.i]]

+ (1 - [[sigma].sub.i])ln [[p.sup.H].sub.i] + ln E([PI.sub.i])

- ln q([[gamma].sub.i], [[omega].sub.i], [T.sub.i]) + ln [[alpha].sub.i].

Equation (13) can be used directly for estimation purposes with the following adjustments. First, we "linearize" the precautionary term, setting ln q([[gamma].sub.i], [[omega].sub.i], [T.sub.i]) = [[gamma].sub.i] + [[omega].sub.i][[gamma].sub.i] + [T.sub.i]. Our choice between log-linear or linear variable specification is dictated by the ease of interpretation of the regression coefficient for the variable in question. The interaction between prudence and variability appears to be reasonable intuitively and is motivated in part by the constant relative risk aversion (CRRA) utility form discussed in the appendix. Second, we take the parameters [[sigma].sub.i] (elasticity of substitution between housing and consumption) and [[delta].sub.i] (maintenance expenses) to be constant across individuals; variations in [[delta].sub.i] (related to differences in ability for housing maintenance and repair) may be captured by a zero-mean random error [[epsilon].sub.i]. Third, we approximate differences in [[alpha].sub.i] (utility weight of housing) by available household characteristics like household size ([HS.sub.i]) and age. Equation (13) then becomes, in regression format [5]

(14) ln ([[p.sup.H].sub.i] [H.sub.i])

= [b.sub.0] + [b.sub.1] ln [[p.sup.H].sub.i]

+ [b.sub.2] ln E([PI.sub.i]) + [b.sub.3][[gamma].sub.i]

+ [b.sub.4][T.sub.i] + [b.sub.5][HS.sub.i] + [b.sub.6][[omega].sub.i][[gamma].sub.i]

+ [[epsilon].sub.i].

Our hypotheses are simply [b.sub.1] [less than] 1 (implying that the intraperiod demand elasticity for housing, [sigma], exceeds 0); [b.sub.2] [greater than] 0, or [b.sub.2] = 1, the first being a weak test of the LCPI hypothesis, the second being a test for the narrowly-defined version of the LCPI hypothesis as derived in our theory section; [b.sub.3] [less than] 0, implying that a precautionary motive for postponing consumption is present; and [b.sub.6] [less than] 0, testing the hypothesis that more prudence will imply the choice of a smaller house. Other than [b.sub.0] (which we expect to be positive), the coefficients are not theoretically constrained; in particular the effect of age on the home purchase consists of both a taste effect and a precautionary effect, which cannot be separately identified.

Variable Definitions

Before we present our regression results, we define the variables used in regression equation (14). The endogenous house value variable [[p.sup.H].sub.i] [H.sub.i] is given by the price of the house (the principal residence in all cases) at the time of purchase by household i. The price index [[p.sup.H].sub.i] is calculated from the MIRS data. Each observation is differentiated by region (northeast, north central, south, and west) and by month and is normalized by the west regional average house price for January 1988, the first month of the sample period. (The same method is used by Brueckner and Follain [1989]).

To measure E([PI.sub.i]), we set time of retirement in equation (8) at R = 65; our measure for each household of initial wealth [W.sub.0] is provided by the down payment, [D.sub.i], for the house purchase. While not a perfect measure, it is based on the actual market transaction of the house purchase as reported by the realtor. [6]. We further set s (the time until resolution of the income uncertainty) = 5 and r (the discount rate) = 0.03 (similar to Caballero [1990], who sets r = 0.04, and Hubbard, Skinner and Zeldes [1995], who set r = 0.03) in equation (8) and E([eta]) (the expected proportionate value of income upon realization of the income uncertainty) = 1 to obtain expected permanent income. [7]

The income variable [Y.sub.i] used in the permanent income measure includes noninterest family income after taxes earned before the purchase of the house; it does not include an imputed rental income. Income is not broken down in components; no wealth or tax information is available. Taxes are proxied based on the tax brackets provided in the Statistical Abstract for the relevant years. Income is stratified and given in ten brackets ranging from less than $25,000 to $100,000 or more. For the interior brackets, income is calculated as the midpoint of the bracket. Income of less than 25,000 is set to 20,000 and income greater than 100,000 is set to 120,000--20% less and more, respectively, than the extreme points of the brackets. This selection was tested in our various specifications, but income dummies inserted for the lowest and highest income bracket were found to be insignificant.

We use [[gamma].sub.i], the coefficient of variation in permanent income to help represent q (Caballero [1990] also uses the coefficient of variation to measure permanent income risk). Employing the definition of the coefficient of variation and equation (8), we measure [[gamma].sub.i] as

(15) [[gamma].sub.i] = [[sigma].sub.[eta]][lgroup][frac{([y.sub.i]/r)[[(1 + r).sup.R-s] - 1]}{[(1 + r).sup.R]}][rgroup]

x [[lgroup][D.sub.i] + [frac{([Y.sub.i]/r)[[(1 + r).sup.R] - 1]}{[(1 + r).sup.R]}][rgroup].sup.-1],

treating [[sigma].sub.[eta]] as constant across individuals and using the same parameter values as for our measure of E([PI.sub.i]). Our data limitations imply that no direct measure for income uncertainty is available. In employing the expression for the coefficient of variation of lifetime income in equation (15), we thus must keep [[sigma].sub.[eta]] constant across individuals. Cross-sectional differences in the coefficient of variation arise solely from differences in household wealth relative to income and age.

The variable [T.sub.i] captures age variation in the precautionary variable [q.sub.i]([[gamma].sub.i], [[omega].sub.i], [T.sub.i]) and age effects on the utility weight on housing [[alpha].sub.i]. Age is given in eight brackets where, as with the income variable, we use the midpoint, except for the group under 25 which we set to 22 (about the midpoint between 18 and 25), and the group over 65 years, which is very small and is thrown out. We obtain (expected) time of death [T.sub.i] as the actuarial life expectancy conditional on the age of the head of the household. Household-specific taste variation is further approximated using the variable, [HS.sub.i], for household size.

To better control for variation in [q.sub.i] across households, we additionally use, in one of our regressions, the unexplained part of the choice of mortgage type, fixed-rate mortgage (FRM) or adjustable rate mortgage (ARM), as a measure of consumer prudence [[omega].sub.i] (where we tentatively presume a negative correlation between the choice of an ARM and prudence). We calculate this variable based on a 0, 1 designation of FRM (assigned one) versus ARM (assigned zero). We run a probit regression explaining mortgage type from the same variables that explain house value (ignoring, of course, the [[omega].sub.i] variable) but, as a price variable, replacing the housing price index by the difference between the relevant fixed mortgage rate and variable mortgage rate. The error in this probit regression is used as our proxy for [[omega].sub.i].

In selecting the sample to exclude constrained households we define constraints in standard fashion as follows. The loan-to-value (LTV) ratio is defined as the ratio of the mortgage loan ([[p.sup.H].sub.i][H.sub.i] - [D.sub.i]) to the purchase price of the house ([[p.sup.H].sub.i][H.sub.i]). The payment-to-income (PTI) ratio is defined as the ratio of the mortgage payment (one over the term of the mortgage plus the mortgage rate time the mortgage loan) to income ([Y.sub.i]). [8]

Summary statistics for our data, providing means and standard variations of key variables, are provided in Table I.

IV. RESULTS

Unconstrained Households

To evaluate the precautionary-motive-extended LCPI, we focus first on those households that are guaranteed not to be subject to LTV or PTI constraints. These consumers have two characteristics that make them likely to be proper LCPI consumers: (1) the amount spent on the home purchase is not influenced by the bank; (2) current and anticipated future liquidity considerations must be minor, since these households could have obtained a larger (mortgage) loan from the bank or lowered their down payment and would qualify for a home equity loan if future liquidity needs would arise.

We consider those households with a PTI [geq] 0.28 or an LTV [geq] 0.95 as constrained. [9] (Although some banks maintained tighter constraints in the 1980s, it is safe to say that households then could easily find a bank that employed the milder constraints). However, we also consider the tighter constraints of PTI [geq] 0.25 and LTV [geq] 0.80. The latter is relevant because a down payment of 20%, implying LTV = 0.80, allows the household to avoid paying private mortgage insurance. Defined as unconstrained is any household purchasing a home of less than 95% of the maximum value allowed subject to the LTV and PTI constraints. Other authors, such as Hayashi [1985] and Jones [1990], have sometimes used tighter criteria to weed out potentially constrained households, but the constraints we consider here appear to be sufficient, and tighter constraints would remove too many sample points. For the first set of PTI and LTV constraints, 1,605 unconstrained households remain in the sample; for the tighter PTI and L TV constraints, 616 unconstrained households remain.

For both sets of constraints, regression results by maximum likelihood are corrected for selection bias resulting from truncation of the endogenous house value. The results are presented in Table II, columns 1 and 2. [10] For the larger sample (fewer households are excluded as being constrained), the price elasticity, [b.sub.1] - 1, is as expected: a point estimate equal to - 0.42, significantly different both from 0 and from 1. This is of similar magnitude compared to the results obtained for cross-sectional data by Brueckner and Follain [1989], Harrington [1989], and Sacher [1993] of -0.52, -0.42, and -0.72, respectively. The permanent income coefficient equals 0.94, is highly significant and positive as expected, further, it does not significantly differ from 1 (at the 5% level) as suggested by the model. The precautionary variable is significantly negative, that is, a higher coefficient of variation of permanent income significantly lowers current consumption (and raises precautionary savings). The resul ts for the smaller sample (retaining only those households that were clearly liquid enough to be able to avoid purchasing private mortgage insurance--house price less than 95% of the value allowed subject to the LTV [less than] 0.80 and PTI [less than] 0.25 constraints)--are qualitatively similar as seen by comparing column 2 to column 1.

Thus, the strict version of the LCPI must be rejected in favor of the model with a precautionary savings motive as is consistent with the numerical results of Zeldes [1989], Caballero [1990], Hubbard, Skinner, and Zeldes [1995], and others. The finding of a precautionary motive is also consistent with the empirical results of Haurin and Gill [1987] in the context of the demand for housing, who find that housing consumption falls with the level of income uncertainty. Our results, however, are not consistent with the results of Haurin [1991], who, with data more suitable for measuring income uncertainty, does not detect an effect of income uncertainty on housing consumption.

Next, we examine if liquidity constraints affect the results. We compare the results for the samples of unconstrained households (1,605 and 616 households) to the results for the full sample of 2,364 households. (Note that we will be able to directly compare constrained and unconstrained households at a later point, but that the current comparison is valuable as it is hard to establish which households are truly constrained). Simple ordinary least squares (OLS) regression produces the results displayed in Table II, column 3. These results again support the extended LCPI and are surprisingly similar to the results for the unconstrained households: a similar estimated price elasticity, a permanent income coefficient that is not significantly different from 1, and a significant precautionary effect.

The importance of the precautionary variable in all three of the (sub)samples provides a potential explanation for the result in Jones [1990] that net wealth affects consumption positively for given income. In our view, higher net wealth relative to current income means that a larger share of lifetime income is certain. Thus, precautionary demand for savings falls--current consumption is higher. Jones attributes the positive effect of net wealth on current consumption (for given lifetime income) to a liquidity effect: higher net wealth relaxes a liquidity constraint so that current consumption can be higher. In the next section, we will be able to better evaluate Jones's explanation relative to ours by examining directly the behavior of constrained households relative to unconstrained households.

One of the limitations of the LCPI as applied to housing demand is that it ignores the portfolio implications of the mortgage loan and the home as an asset. These portfolio implications in the demand for housing, and other illiquid durable consumption goods, are examined for instance by Henderson and Ioannides [1983], Plaut (1987] and Grossman and Laroque [1990]. To the extent that younger households who are first-time home buyers typically have too few nonhousing assets to be seriously concerned about the role of house and mortgage loan in their portfolios, our model is more likely to apply to young households. [11]

Thus, to check the robustness of our results, we further limit the unconstrained household sample by excluding all households with head older than 34 years and those who are not first-time home buyers. (Thirty-four years is the typical cut-off for young versus old households; see, for instance, Jones [1990], Duca and Rosenthal [1993], or Sheiner [1995]). This reduces the sample of the unconstrained from 1,605 to 352. The results for this subsample are displayed in Table II, column 4. These results are very similar to our previous results with the proviso that the precautionary effect now becomes insignificant, although its coefficient value remains similar.

Before considering constrained households, we take advantage of the specifics of our data set to use the unexplained part of the choice of mortgage type as a proxy for [[omega].sub.i], household prudence, where home owners with unexplained preference for ARMs are tentatively identified as less cautious. Column 5 in Table II displays the results when a "prudence" variable is added to the basic regression of column 1. This variable is defined as the product of the coefficient of variation of permanent income ([[gamma].sub.i]) times the unexplained part of the adjustable versus fixed rate mortgage choice (which proxies for [[omega].sub.i]). It is expected to have a negative impact on house value, if households with an unexplained preference for FRMs are more cautious.

The results in column 5 are consistent with the results in column 1. Although the prudence variable has a negative effect in column 5, the coefficient is not significant. Possible explanations for the lack of significance are that (a) for certain households FRM contracts may in fact be riskier than ARM contracts, as discussed in Szerb [1996]; or that (b) the ARM choice is less indicative of lack of caution than it is indicative of the household's expectation to move in a relatively short time. Specifically, a household expecting to move quickly is more likely to take the ARM, which typically has a rate that is initially lower than the FRM rate; but such a household may also be more likely to buy a smaller house. In this case, the "mobility" effect may offset the "prudence" effect. The mobility effect may be less relevant in our sample, where households acquire the mortgage when they buy the house, since it is rarely optimal to buy a new home if one intends to move shortly thereafter.

In further robustness checks, we replaced our permanent income variable with income and with the down payment. In both cases, the results deteriorated. We also added income and the down payment separately to the regression including the permanent income variable, but this led to serious multi-collinearity problems. These results are not presented.

Liquidity-Constrained Households and an Endogenous Switching Approach

In the housing literature, recent work by Sheiner [1995] shows that liquidity constraints of renters are an important factor in tenure choice, whereas Duca and Rosenthal [1993] report that 30% of young households in the Survey of Consumer Finances consider themselves liquidity constrained. We consider the housing choice of new homeowners that are likely to be liquidity constrained based on the realtor-provided information in our data set.

Although it is fairly straightforward in our data set to select households that are definitely unconstrained (we essentially observe excess liquidity), it is much more difficult to identify definitely constrained households (we cannot observe a "shortage" of liquidity; for instance, due to possible government guarantees or the possibility that not all current wealth was used as down payment). We thus consider as constrained, fairly arbitrarily, those households that buy a home worth more than 105% of the maximum value allowed by the LTV and PTI constraints. The results of the regressions, again correcting for truncation bias, are presented in Table III, column 6--reporting results for the complement of the data used in column 1 minus all households with homes valued between 95% and 105% of the amount implied by LTV = 0.95 and PTI = 0.28--and column 7 --reporting results for the complement of the data used in column 2 minus all households with homes valued between 95% and 105% of the amount implied by LTV = 0.80 and PTI = 0.25. The results are not much different from the results for the unconstrained households. The main difference is that the permanent income coefficients now are significantly less than i and equal to 0.73 for the smaller sample (565 households) and 0.84 for the larger sample (1,298 households).

The degree of caution is somewhat lower for the constrained households in both sub-samples. If the view of Jones [1990] were correct, then our precautionary variable would proxy for the (inverse of) net wealth and would imply an apparently stronger "precautionary" effect for constrained households. If anything, however, the precautionary variable is less important for constrained households. Thus, the precautionary effect seems to dominate the liquidity effect. Nevertheless, the lower value for the permanent income coefficient in the constrained households case suggests that liquidity does matter to some extent.

To further examine the nature of a possible liquidity effect, we again consider the ARM versus FRM households. Xu [1995] has shown that, in the presence of potential liquidity constraints, two types of precautionary savings should occur: one that anticipates potential negative adjustments to lifetime wealth and one that anticipates potential liquidity shortages. Although holders of ARMs need not be less cautious in terms of lifetime wealth as we found earlier (and as emphasized theoretically by Szerb [1996]: ARMs tend to be less risky than FRMs if inflation premia are more variable than real interest components), holders of ARMs are definitely less cautious in terms of liquidity problems. A household facing liquidity shortages would do well to consider the FRM, in spite of the typically lower expected interest costs offered by the ARM, to avoid potential increases in interest payments that would exhaust liquidity. If a liquidity-constrained household nevertheless chooses the ARM, this suggests a low degree o f caution.

If liquidity considerations are important the above discussion suggests that liquidity constrained households that are ARM holders have a low degree of caution (as discussed earlier: if they are just more likely to move soon, why would they purchase a new home at this point in time?). They therefore should respond less to changes in the variation of their lifetime income than should liquidity-constrained FRM holders. The regressions for constrained households presented in Table III, columns 6 and 7, include an interaction variable for the coefficient of variation of permanent income (CFVAR) and the mortgage choice (ARM choice equals 0) as in column 5, Table II. The results reveal that the interaction variable is insignificant. This further supports the view, from the perspective of the households at least, that liquidity considerations are not crucial.

Typically, households that are younger are more likely to be liquidity constrained. This is one reason that many consumption studies have been limited to young households. To a large extent, our selection procedure for constrained households is more precise than a pure age-based criterion, since it looks at income and wealth variables that, in principle, directly determine liquidity. Given, however, that our measures are imperfect, age may be helpful as an additional sample selection criterion in identifying constrained households. Thus, we further restrict the sample based on age, in this case by retaining from the subsample in column 7 only those constrained households who are first-time home buyers and whose head is 34 years or younger, reducing sample size from 1,298 to 553. This selection criterion again also has the advantage of limiting the sample to young households, who are less likely to choose house and mortgage based on portfolio considerations and for whom therefore the down payment is more like ly to be close to current wealth. The results are displayed in column 8. Results are similar to those in columns 6 and 7 except that the permanent income coefficient is now 0.87, a little closer to i. Further, the precautionary variable becomes insignificant as is the case for the age-restricted sample of unconstrained households in column 4, Table II.. It appears that the basic LCPI model may indeed better describe the behavior of young households.

Although households may not base consumption choices on current or expected future liquidity positions, banks have to deal with information asymmetries and may take LTV and PTI constraints seriously. From this perspective, a household chooses the value of the house when unconstrained, whereas a bank specifies the value of the house when the household is constrained. One may then think of the regression model as one of endogenous switching, where the outcome is the optimal consumer choice given in equation 13 if no constraint is binding and the outcome is the constrained value (set by the bank) when either the LTV constraint or the PTI constraint is binding.

The value set by the bank is modeled as

(16) ln[([[p.sup.H].sub.i][H.sub.i]).sub.bank]

= [c.sub.0] + [c.sub.1] ln([Hmax.sub.i]) + [[nu].sub.i],

where the Hmax variable indicates the highest value of the house allowed by the bank given the LTV constraint of 0.95 and the PTI constraint of 0.28. The error term is assumed to be white noise. It would be expected that [c.sub.0] = 0 and that [c.sub.1] = 1. The value of the house observed equals the minimum of the left-hand side of equation (14) and the left-hand side of equation (16). This model then is a standard "disequilibrium" model as described by Maddala [1983, 2961.

Estimation of the model by maximum likelihood would not converge. We accordingly employ the suggestion of Goldfeld and Quandt [1975] to impose a restriction among the standard errors of the separate regressions. As the standard errors of both equations estimated by OLS are approximately equal we impose this equality restriction. The results are presented in Table III, column 9. The result for the constrained households is that the coefficient [c.sub.0] is significantly positive, whereas [c.sub.1] = 0.43, which is significantly positive but less than 1. Thus, banks have a significant but limited impact on the housing choice of constrained households: a 1% decrease in the maximum house value allowed by the bank based on PTI and LTV considerations lowers the value of the purchased house, but only by 0.43%. The parameter estimates for the unconstrained households are similar to our results in column 1, but the permanent income elasticity is now equal to 1.24 and significantly above 1.

For reasons discussed above, we again limit the sample to first-time home buyers with head age 34 or younger, reducing the sample from 2,364 to 703 households. Column 10 mostly confirms the results of column 9. The difference is that the permanent income coefficient is now no longer significantly different from i. As is the case with the previous age-restricted regressions, and may be related to the reduced sample size, the precautionary variable becomes insignificant but retains a value that is consistent with the precautionary variable coefficients of the non-age-restricted regressions. Thus, again the age restriction appears to strengthen the case for the LCPI (without liquidity constraints).

A possible explanation for the constrained household results may be related to the fact that banks, in the case of constrained households, do have some effect on the value of the house purchased as follows from columns 9 and 10. This could explain the lower permanent income elasticity for the constrained households found in columns 6 and 7. Since the results in columns 6 and 7 fail to identify a household-initiated concern for future liquidity, it follows, with the proviso that our data limitations may have introduced biases, that the results in Table II, columns 1 and 2, reveal a true precautionary savings effect on current consumption related to variability in lifetime income, that is unrelated to liquidity considerations.

V. CONCLUSION

Households that face no current liquidity constraints and should have relatively costless access to liquidity in the future, make housing choices in accordance with an extended version of the LCPI that allows for precautionary savings. As such, we add to the results of Jones [1990], Bernanke [1984], and Hayashi [1985] by focusing initially only on non-liquidity-constrained consumers in the "Unconstrained Households" subsection of Section IV so that the precautionary and liquidity effects may be separated.

Our results allow a reinterpretation of the Jones [1990] observation that net wealth matters separately from lifetime income. He concludes that the importance of net wealth points to liquidity constraints. However, we note that higher net wealth also implies less permanent income variability, so that the importance of net wealth could point to precautionary motives just as easily as to liquidity constraints. We provide some evidence for the precautionary motive and preliminarily distinguish the precautionary motive explanation from the liquidity constraint explanation by providing additional evidence against the importance of household-initiated liquidity effects (based on the absence of differences between unconstrained and liquidity-constrained borrowers, and between liquidity-constrained ARM and FRM holders, regarding the income uncertainty variable). We further find that the minor liquidity problems that appear to exist may be bank initiated, as a way to mitigate the effects of informational asymmetries, rather than household initiated. A conclusion that liquidity constraints have only a minor impact on housing choice is further supported by the fact that banks seem to pay only limited attention to the PTI and LTV indicators of liquidity constraints.

Our results suggest that precautionary savings may explain results previously attributed to liquidity constraints. These results can only be preliminary due to data limitations. In particular, the lack of direct observations on income variability prohibits a definitive separation of liquidity and precautionary effects. Given that Haurin [1991], with more appropriate data, did not identify an effect of income uncertainty on housing demand, further work is necessary to obtain clarity on this issue. In addition, the problem of approximating wealth with the down payment may invalidate our results. Restricting the sample to young first-time home buyers deals in part with this problem, but the precautionary effect now becomes insignificant, likely due to small sample size. Finally, the restriction of our sample to home owners alone may provide a bias that is only partly corrected by truncation of housing values below a threshold value and then correcting for the known bias.

Although our data are limited in these dimensions, the results at least are robust in the face of attempts to correct biases; they further allow a look at the life cycle hypothesis from a novel perspective. Future work may apply panel data from the Survey of Consumer Finances in the context of our model to obtain further evidence that may help to separate the precautionary motive from liquidity constraint effects.

(*.) The authors thank two anonymous referees for very helpful comments; one referee in particular provided extensive and constructive comments that substantially improved the empirical part of the paper.

Balvers: Professor of Economics, West Virginia University, Morgantown

Szerb: Associate Professor, Janus Pannonius University, P[acute{e}]cs, Hungary.

(1.) This is not to say that earlier work in the housing literature has not addressed the precautionary motive. In particular, Plaut [1987] considers theoretically the precautionary savings that help absorb the risk inherent in owning a house; Haurin and Gill [1987] and Haurin [1991] consider empirically the effect of income uncertainty on housing consumption.

(2.) Although lifetime should typically be viewed as stochastic as well, it would be possible given perfect insurance markets to buy an actuarially fair annuity that guarantees the level of consumption derived here but for the stochastic lifetime (see also Caballero [1991, 870] for a similar argument). All decisions then would be identical. Note that income insurance is a very different issue because severe moral hazard and adverse selection problems make a perfect market for such insurance highly implausible.

(3.) It is easy to check that utility can be increased for a given budget if consumption is not smoothed perfectly before the income uncertainty is resolved. A similar argument applies to guarantee perfect consumption smoothing after the income uncertainty is resolved.

(4.) Notice that for r [rightarrow] 0 the use of L'H[hat{o}]pital's rule implies that PI = [[W.sub.0] + sY + (R - s)[eta]Y]/T.

(5.) We have experimented with including the following additional variables in equation (14): an unemployment rate variable (varying by region and by year) obtained from the Current Population Survey to better capture income uncertainty. This variable turned out to be insignificant, often with the wrong (positive) sign, in all regressions. We also considered the square of age as an additional variable to capture life cycle effects. This variable was usually significantly negative as expected but did not have an important effect on the coefficients of the other variables, and is not included. Regional dummies were considered but then dropped from our regressions, since their explanatory power is fully subsumed by the price variable.

(6.) Skinner [1994] reports that housing equity for the median household is around 60% of total current wealth. Assuming that, for the typical household, current wealth accumulates after the initial down payment, the approximation of current wealth at the time of the home purchase by the down payment appears to be reasonable. The results of Duca and Rosenthal [1993] imply that current wealth and debt would be positively correlated. Our assumption here, equating the down payment with current wealth, implies that, for a given house size, current wealth and mortgage debt are negatively correlated. However, since current wealth is positively correlated with lifetime income, higher current wealth also implies a more expensive house purchase so that mortgage debt may be positively related to current wealth in our model as well.

(7.) These rather arbitrary parameter choices have little effect on the empirical results. Checking the robustness of the main model we use r = .02 and r = .04, s = 3 and s = 7 instead and find no qualitative changes in our results. Similarly we set initial wealth at twice the down payment (instead of equal to the down payment) and find no qualitative changes in our results.

(8.) We calculate the mortgage rate in the PTI constraint using the FRM rate (varying only by month and by region) for all households. This avoids the endogeneity problem arising if households choose the ARM, with, typically, an initially lower rate, to avoid becoming constrained. Results when the actual rate paid by the household is used in calculating the PTI are not presented but are similar in all respects to the results presented below. Similarly, in our probit regression for the prudence proxy, we use average FRM and ARM rates (varying only by month and by region) to avoid the problem that, for individual households, only the actual rate for the chosen mortgage type is available.

(9.) As discussed in Phillips and Vanderhoff [1994, 459], banks require a maximum LTV ratio that is in all cases 0.90 or higher. Banks also require a PTI ratio of no more than 0.28. See for instance Brueckner and Follain [1989].

(10.) The fact that no renters are included in the sample causes a potential sample selection bias that, as for example, in Jones [1990], we cannot easily correct. Households desiring small homes may be more likely to rent and, if their housing demand has a negative "error" component, are less likely to be in the sample. As this is more likely to occur at low permanent income levels, the estimate of the permanent income elasticity may be biased downward. One, imperfect, way to correct for the bias is to truncate the sample to eliminate all low house values and then correct for the truncation bias. When we do this in the regressions of columns 1 and 2 for threshold house values of $50,000 and $80,000, we find very little impact on the coefficient estimates; the permanent income coefficients are higher in all cases but negligibly so. Thus, it appears that the sample selection bias due to absence of renters is minor.

(11.) As discussed by Henderson and loannides [1983], portfolio considerations contribute to households having different consumption and investment demands for housing. Observed housing may therefore not reflect a consumption demand for housing. Ioannides and Rosenthal [1994] show however that, empirically, the residence of most homeowners is determined primarily by their consumption demand.

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 Summary Statistics
 All Households Unconstrained
 Standard Standard
Variable Mean Deviation Mean Deviation
House value 125,221 89,478 141,992 87,920
 (dollars)
Down payment 29,627 44,243 57,636 59,727
 (dollars)
After-tax income 40,912 18,690 46,234 19,774
 (dollars per annum)
Permanent income 32,934 15,248 36,276 16,532
 (dollars per annum)
Coefficient of variation 0.69 0.17 0.62 0.21
 permanent income
Age 36.0 9.1 40.2 9.59
 (years)
Household size 2.76 0.98 2.76 0.80
 (units)
Loan-to-value 0.81 0.16 0.62 0.14
 (fraction)
Payment-to-income 0.18 0.07 0.14 0.05
 (fraction)
Mortgage rate 9.84 0.73 9.90 0.68
 (percentage)
Sample size 2,364 -- 616 --
 (units)
 ARM Holders
 Standard
Variable Mean Deviation
House value 158,470 115,185
 (dollars)
Down payment 37,572 49,429
 (dollars)
After-tax income 43,887 20,151
 (dollars per annum)
Permanent income 35,726 16,570
 (dollars per annum)
Coefficient of variation 0.69 0.16
 permanent income
Age 36.0 8.8
 (years)
Household size 2.66 1.08
 (units)
Loan-to-value 0.80 0.13
 (fration)
Payment-to-income 0.18 0.08
 (fraction)
Mortgage rate 8.65 0.68
 (percentage)
Sample size 457 --
 (units)
Note: Unconstrained households are defined as those with a house
value less than 95% of the value obtainable given a maximum loan-
to-value ratio of 0.80 and a maximum payment-to-income ratio of 0.25.
 Unconstrained Households
Variable 1 2 3 4
Constant 3.76 4.01 2.26 3.68
 (7.68) (4.38) (13.30) (2.10)
Log of price index 0.59 0.63 0.45 0.33
[[p.sup.H].sub.i] (6.54) (3.46) (14.68) (3.43)
Log of permanent income 0.94 0.92 1.02 0.91
E([PI.sub.i]) (20.72) (10.81) (60.11) (17.47)
PI variability -1.37 -0.98 -1.42 -1.25
[[gamma].sub.i] (7.29) (3.06) (17.90) (1.71)
Age -0.01 -0.01 -0.00 -0.03
[T.sub.i] (3.02) (1.60) (2.11) (2.89)
Household size -0.03 -0.01 -0.03 -0.02
[HS.sub.i] (1.58) (0.11) (3.80) (1.08)
(Mortgage type error) * -- -- -- --
 (PI Var.) [[omega].sub.i][[gamma].sub.i]
Average house size 126 142 125 97
[[p.sup.H].sub.i][H.sub.i]
Sample size 1605 616 2364 352
Variable 5
Constant 3.76
 (7.80)
Log of price index 0.58
[[p.sup.H].sub.i] (6.41)
Log of permanent income 0.94
E([PI.sub.i]) (20.68)
PI variability -1.29
[[gamma].sub.i] (6.53)
Age -0.01
[T.sub.i] (2.99)
Household size -0.03
[HS.sub.i] (1.53)
(Mortgage type error) * -0.10
 (PI Var.) [[omega].sub.i][[gamma].sub.i] (1.43)
Average house size 126
[[p.sup.H].sub.i][H.sub.i]
Sample size 1605


Notes: Absolute t-statistics in parentheses. The independent variable is in all cases the log of the purchase price of the house. All equations except for column 3 are estimated by maximum likelihood corrected for truncation bias using the Newton method for computing standard errors. Column 1 displays the regression result based on equation (15) for the sample of unconstrained households whose house value is less than 95% of the maximum feasible given LTV [less than] 0.95 and PTI [less than] 0.28. Column 2 displays the result for the sample of unconstrained households with house value less than 95% of the maximum feasible given LTV [less than] 0.80 and PTI [less than] 0.25. Column 3 displays the OLS regression results for the full sample. Column 4 considers the sample in column 1 further restricted to exclude households with head age exceeding 34 years and those who are not first-time home buyers. In column 5, the interaction dummy variable for mortgage choice (ARM = 0) with permanent income uncertainty is a dded for the sample in column 1.
 Constrained Households
Variable 6 7 8 9 10
Constant 4.48 3.47 1.10 0.84 3.83
 (7.80) (12.38) (0.96) (2.81) (2.89)
Log of price index 0.32 0.32 0.30 0.67 0.36
 [[p.sup.H].sub.i] (4.00) (6.81) (5.38) (12.64) (6.66)
Log of permanent income 0.73 0.84 0.87 1.24 0.95
 E([PI.sub.i]) (12.56) (30.18) (24.48) (36.95) (25.45)
PI variability -0.82 -0.84 -0.97 -1.86 -1.45
 [[gamma].sub.i] (2.95) (6.89) (0.80) (13.64) (1.09)
Age [T.sub.i] -0.00 -0.00 -0.03 -0.01 -0.04
 (0.01) (0.36) (5.71) (3.48) (6.67)
Household size [HS.sub.i] -0.03 -0.02 -0.03 -0.04 -0.04
 (1.64) (2.00) (2.34) (3.63) (3.33)
(Mortgage type error)* -0.12 -0.05 -0.04 -- --
(PI Var.) [[omega].sub.i] (1.33) (1.29) (0.93)
[[gamma].sub.i]
Constant [c.sub.0] -- -- -- 6.85 8.21
 (47.28) (29.51)
Bank max [c.sub.1] -- -- -- 0.43 0.33
 (35.09) (13.52)
Average house size 114 112 87 125 91
 [[p.sup.H].sub.i][H.sub.i]
Sample size 565 1,298 553 2,364 703


Notes: The independent variable is in all eases the log of the purchase price of the house. Equations are estimated by maximum likelihood. Columns 6, 7, and 8 are corrected for truncation bias using the Newton method for computing standard errors. Columns 6 and 7 display the regression results based on equation (15) for the sample of constrained households (with house price larger than 105% of the maximum value, given LTV [geq] 0.80 and PTI [geq] 0.25, and LTV [geq] 0.95 and PTI [geq] 0.28, respectively). Column 8 represents the same regression as in column 7 with the sample further limited to households with head age less than 35 who are first-time home buyers. Columns 9 and 10 use the endogenous switching likelihood function as given by Maddala [1983, eq. (10.21)] and present the results based on equations (15) and (16), for the full sample in column 9 and, for first-time home buyers under 35 years old, in column 10.

ABBREVIATIONS

ARM: adjustable-rate mortgage

CES: constant elasticity by substitution

CFVR: coefficient of variation of permanent income

CRRA: constant relative risk aversion

FRM: fixed-rate mortgage

LCPI: life cycle/permanent income [hypothesis]

LTV: loan to value [ratio]

MIRS: Mortgage Insurance Rate Survey

OLS: ordinary least squares

PTI: payment to income [ratio]

APPENDIX

The Determinants of Precautionary Savings

Taking expectations on both sides of equation (8) yields

(Al) Z = E(PI)/q,

where q [equiv] 1 + f(T)E(Q) and f(T) [equiv] [[(1 + r).sup.T-s] - 1]/[[(1 + r).sup.T] - 1]. Since PI = E(PI) + [epsilon] we can write by definition of the coefficient of variation, [gamma], that

(A2) PI = (1 + [gamma][kappa])E(PI),

where [kappa] is a standardized random variable with mean zero and variance of one. Substitute equations (8), (Al) and (A2) into equation (12) to obtain

(A3) U'[lgroup][frac{E(PI)}{q}][rgroup]

= E{U'[E(PI)[lgroup][frac{f(T) - 1 + q}{f(T)q}] + [frac{[gamma][kappa]}{f(T)}][rgroup]]}.

From (A3) we thus have in general that q = q[[gamma], E(PI), T, [omega]], where [omega] is a shift variable of the utility function U, but we also want to establish the direction of the effect that these variables have on q.

An increase in [gamma] (all else equal) implies a mean preserving spread of the function argument on the right-hand side of (A3). Under "prudence" U'( ) is convex so that the r.h.s. of (A3) increases. Thus the l.h.s. should increase as well implying that q increases, since U( ) is concave, so that [q.sub.[gamma]] [greater than] 0. Similarly, an increase in prudence ([omega] increases) raises q, [q.sub.[omega]] [greater than] 0.

The effects of E(PI) and T are more complex. To examine these, we assume constant relative risk aversion for the utility function (which implies prudence). It is clear from (A3) that E(PI) drops from both sides if CRRA is assumed. Thus, [q.sub.E(PI)] = 0. Even under CRRA the effect of T cannot be signed: [q.sub.T] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are stated in the text.
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